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Theorem shftlem 12881
Description: Two ways to write a shifted set  ( B  +  A ). (Contributed by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
shftlem  |-  ( ( A  e.  CC  /\  B  C_  CC )  ->  { x  e.  CC  |  ( x  -  A )  e.  B }  =  { x  |  E. y  e.  B  x  =  ( y  +  A ) } )
Distinct variable groups:    x, y, A    x, B, y

Proof of Theorem shftlem
StepHypRef Expression
1 df-rab 2826 . 2  |-  { x  e.  CC  |  ( x  -  A )  e.  B }  =  {
x  |  ( x  e.  CC  /\  (
x  -  A )  e.  B ) }
2 npcan 9841 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  A  e.  CC )  ->  ( ( x  -  A )  +  A
)  =  x )
32ancoms 453 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( ( x  -  A )  +  A
)  =  x )
43eqcomd 2475 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  x  =  ( ( x  -  A )  +  A ) )
5 oveq1 6302 . . . . . . . . . 10  |-  ( y  =  ( x  -  A )  ->  (
y  +  A )  =  ( ( x  -  A )  +  A ) )
65eqeq2d 2481 . . . . . . . . 9  |-  ( y  =  ( x  -  A )  ->  (
x  =  ( y  +  A )  <->  x  =  ( ( x  -  A )  +  A
) ) )
76rspcev 3219 . . . . . . . 8  |-  ( ( ( x  -  A
)  e.  B  /\  x  =  ( (
x  -  A )  +  A ) )  ->  E. y  e.  B  x  =  ( y  +  A ) )
87expcom 435 . . . . . . 7  |-  ( x  =  ( ( x  -  A )  +  A )  ->  (
( x  -  A
)  e.  B  ->  E. y  e.  B  x  =  ( y  +  A ) ) )
94, 8syl 16 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( ( x  -  A )  e.  B  ->  E. y  e.  B  x  =  ( y  +  A ) ) )
109expimpd 603 . . . . 5  |-  ( A  e.  CC  ->  (
( x  e.  CC  /\  ( x  -  A
)  e.  B )  ->  E. y  e.  B  x  =  ( y  +  A ) ) )
1110adantr 465 . . . 4  |-  ( ( A  e.  CC  /\  B  C_  CC )  -> 
( ( x  e.  CC  /\  ( x  -  A )  e.  B )  ->  E. y  e.  B  x  =  ( y  +  A
) ) )
12 ssel2 3504 . . . . . . . . . 10  |-  ( ( B  C_  CC  /\  y  e.  B )  ->  y  e.  CC )
13 addcl 9586 . . . . . . . . . 10  |-  ( ( y  e.  CC  /\  A  e.  CC )  ->  ( y  +  A
)  e.  CC )
1412, 13sylan 471 . . . . . . . . 9  |-  ( ( ( B  C_  CC  /\  y  e.  B )  /\  A  e.  CC )  ->  ( y  +  A )  e.  CC )
15 pncan 9838 . . . . . . . . . . 11  |-  ( ( y  e.  CC  /\  A  e.  CC )  ->  ( ( y  +  A )  -  A
)  =  y )
1612, 15sylan 471 . . . . . . . . . 10  |-  ( ( ( B  C_  CC  /\  y  e.  B )  /\  A  e.  CC )  ->  ( ( y  +  A )  -  A )  =  y )
17 simplr 754 . . . . . . . . . 10  |-  ( ( ( B  C_  CC  /\  y  e.  B )  /\  A  e.  CC )  ->  y  e.  B
)
1816, 17eqeltrd 2555 . . . . . . . . 9  |-  ( ( ( B  C_  CC  /\  y  e.  B )  /\  A  e.  CC )  ->  ( ( y  +  A )  -  A )  e.  B
)
1914, 18jca 532 . . . . . . . 8  |-  ( ( ( B  C_  CC  /\  y  e.  B )  /\  A  e.  CC )  ->  ( ( y  +  A )  e.  CC  /\  ( ( y  +  A )  -  A )  e.  B ) )
2019ancoms 453 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( B  C_  CC  /\  y  e.  B )
)  ->  ( (
y  +  A )  e.  CC  /\  (
( y  +  A
)  -  A )  e.  B ) )
2120anassrs 648 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  C_  CC )  /\  y  e.  B
)  ->  ( (
y  +  A )  e.  CC  /\  (
( y  +  A
)  -  A )  e.  B ) )
22 eleq1 2539 . . . . . . 7  |-  ( x  =  ( y  +  A )  ->  (
x  e.  CC  <->  ( y  +  A )  e.  CC ) )
23 oveq1 6302 . . . . . . . 8  |-  ( x  =  ( y  +  A )  ->  (
x  -  A )  =  ( ( y  +  A )  -  A ) )
2423eleq1d 2536 . . . . . . 7  |-  ( x  =  ( y  +  A )  ->  (
( x  -  A
)  e.  B  <->  ( (
y  +  A )  -  A )  e.  B ) )
2522, 24anbi12d 710 . . . . . 6  |-  ( x  =  ( y  +  A )  ->  (
( x  e.  CC  /\  ( x  -  A
)  e.  B )  <-> 
( ( y  +  A )  e.  CC  /\  ( ( y  +  A )  -  A
)  e.  B ) ) )
2621, 25syl5ibrcom 222 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  C_  CC )  /\  y  e.  B
)  ->  ( x  =  ( y  +  A )  ->  (
x  e.  CC  /\  ( x  -  A
)  e.  B ) ) )
2726rexlimdva 2959 . . . 4  |-  ( ( A  e.  CC  /\  B  C_  CC )  -> 
( E. y  e.  B  x  =  ( y  +  A )  ->  ( x  e.  CC  /\  ( x  -  A )  e.  B ) ) )
2811, 27impbid 191 . . 3  |-  ( ( A  e.  CC  /\  B  C_  CC )  -> 
( ( x  e.  CC  /\  ( x  -  A )  e.  B )  <->  E. y  e.  B  x  =  ( y  +  A
) ) )
2928abbidv 2603 . 2  |-  ( ( A  e.  CC  /\  B  C_  CC )  ->  { x  |  (
x  e.  CC  /\  ( x  -  A
)  e.  B ) }  =  { x  |  E. y  e.  B  x  =  ( y  +  A ) } )
301, 29syl5eq 2520 1  |-  ( ( A  e.  CC  /\  B  C_  CC )  ->  { x  e.  CC  |  ( x  -  A )  e.  B }  =  { x  |  E. y  e.  B  x  =  ( y  +  A ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2818   {crab 2821    C_ wss 3481  (class class class)co 6295   CCcc 9502    + caddc 9507    - cmin 9817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-ltxr 9645  df-sub 9819
This theorem is referenced by: (None)
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